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This is a question from an old qualifying exam.

Let $f=u+iv$ be a holomorphic function on the open unit disc into the complex plane. Among all holomorphic functions $f$ with $v(0) = 0$ and $|u|<1$ what is the largest possible value of $v(1/2)$.

Attempt:

My initial idea was to let $g = e^f$ so that $|g(z)| = e^{u(z)}$ and $$r = \frac{z-e^{u(0)}}{e-\frac{1}{e}e^{u(0)}z}.$$ Then letting $h = r\circ g$, which is a holomorphic map from the unit disc into the unit disc with $h(0) = 0$, we can apply Schwarz's Lemma to conclude that $|h(1/2)| \leq 1/2$.

This seems to be a standard way to approach these types of problems, however I wasn't able to find an upper bound on $v(1/2)$ using this method.

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    $\begingroup$ Here is a way to bound $|g(1/2)|$, this implies the bound for $|v(1/2)|$ which I think will be sharp. $\endgroup$ – user100000 Oct 17 '13 at 1:52
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Schwarz lemma states that the equality of $|f(z)|\le |z|$ holds iff $f$ is a rotation. Use this.

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